Electromagnetic Waves. 163 



■ordinary wave-equation expressed in terms of cylindrical 

 •coordinates. 



In the case o£ symmetry about the axis of z, the solutions 

 (7*1) reduce to a solution given by Hertz * ; the function 

 which Hertz calls n being in fact the same as the function 

 — U. It will be remembered that U is now independent 

 of </>, and consequently that Y = 0, a-=0. 



Similarly, still assuming symmetry about the axis of z, the 

 solutions (7'3) represent the field of FitzGerald's magnetic 

 oscillator f ; here V is independent of <j> and X = 0, {3 = 0. 



The solutions (1*6), (1*7) include also a solution given by 

 Lord Rayleigh "f and applied by him to the propagation of 

 electromagnetic waves in the space outside perfect con- 

 ductors ; we present the formulae here in the form given by 

 Macdonald §. We take f, rj as conjugate coordinates in 

 the plane of xy, and identify f with s, (making as above a 

 cyclical interchange of coordinates). Then we have 



A = B = J, = 1, 

 because ds 2 = J 2 (dp + drj 1 ) + dz 2 . 



Lord Rayleigh' s solution is found by assuming that 



Z = 0, 7 = 0, 



so that the electric and magnetic forces both lie in the 

 wave-fronts. 



Then we find the formulae corresponding to (1*6) : 



r 1 b /KJD\ 1 



_ 1 jy^u 

 i yv 



z = o. 



7 = 0, J 



where U is now a solution of the two equations : 



5?" V a? = 0> and W + a? = °- (7 ' 5) 



Lord Rayleigh's actual solutions are given by taking fi = l 9 

 .K = l, and U = ^<^ ± - ?) F(f,i ? ). 



* ' Electric Waves' (English edition), p. 140. 



t ' Scientific Writings,' p. 1:22 : it will be seen that FitzGerald does not 

 give the explicit formulae (7-3) in this paper, although he obtains the 

 corresponding formula for the rate of radiation of energy which can be 

 -derived from (73) in the same way as (4 - 7). In a later paper (/. c. 

 p. 418) the parallelism of the fields (7*1) and (7*3) is pointed out. 



X Phil. Mag. xliv. p. 199 (1897) ; Scientific Papers, vol. iv. p. 327. 



§ 'Electric Waves/ ch. vii. §43. 



M 2 



